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Full text of "Treatise On Analysis Vol-Ii"

8   CONVOLUTION OF A MEASURE AND A FUNCTION        281
But the lemma (14.8.1.1) shows that this is equivalent to the relation

and therefore (13.21.10) equivalent to

f*,/    x      n     x   f%,   -1    x

h(x)dp(x)    f(s    x)dii(s)< +00.
J                J

But this is a consequence of our hypotheses (13.13.1).

When the conditions of (14.8.1) are satisfied, we say that the measure ju and
the function fare convolvable. Any function which is equal almost everywhere
(with respect to /?) to the function g of (14.8.1) is called (by abuse of language)
the convolution of /* and/, and is written ju */(or n * *f if the Haar measure /?
needs to be brought into the notation). Thus we have, almost everywhere with
respect to /?,

(14.8.2)                          (M */)(*) =

If one of the functions equal almost everywhere to g is continuous on G,
then it is the only function with this property, because the support of /? is
equal to G; in this case it is this function which is denoted by \JL */. We then
have, in particular,

(14.8.3)                         (M*/)0) =

Similarly we define / and u to be convolvable if the measures / * j8 and /*
are convolvable. For this it is necessary and sufficient that, for almost all x
(with respect to /?) the function 1sH->/(xs~"1)A($~1) should be /^-integrable and
that the function

J-

should be locally jS-integrable. The measure (/ ft) * \i then has a density with
respect to ft, and we denote any one of these densities by / * u, so that for
almost all x (with respect to /?) we have

(14.8.4)                  (/ * n)(x)  I f(xs~ ^(s" *) d/i(0.se that the condition is satisfied. We have to show that,